Colóquio 

14/05/2013, 10:30 — 11:30 — Sala P3.10, Pavilhão de Matemática
Alessio Figalli, University of Texas at Austin

Stability results for sumsets in \(\mathbb{R}^n\)

Given a Borel set \(A\) in \(\mathbb{R}^n\) of positive measure, one can consider its semisum \(S=(A+A)/2\). It is clear that \(S\) contains \(A\), and it is not difficult to prove that they have the same measure if and only if \(A\) is equal to his convex hull minus a set of measure zero. We now wonder whether this statement is stable: if the measure of \(S\) is close to the one of \(A\), is \(A\) close to his convex hull? More in general, one may consider the semisum of two different sets \(A\) and \(B\), in which case our question corresponds to proving a stability result for the Brunn-Minkowski inequality. When \(n=1\), one can approximate a set with finite unions of intervals to translate the problem onto \(\mathbb{Z}\), and in the discrete setting this question becomes a well studied problem in additive combinatorics, usually known as Freiman's Theorem. In this talk I'll review some results in the one-dimensional discrete setting, and show how to answer to this problem in arbitrary dimension.